The unitary group, $U(n)$, acts transitively on the Grassmann manifold $X = Gr(2, C^n)$. The isotropy group is $H = U(2)\times U(n-2)$, i.e. the group elements leaving some $x$ fixed. What are the dimensions of the orbits of $H$ in $X$? The brute force calculation gets messy for some of the orbits, so I am wondering if these results are published somewhere. The generic orbit is codimension 2, and there are the special orbits of $\{x\}$ and $x_\perp = Gr(2,C^{n-2})$, but there are other orbits, too.